COMSOL Multiphysics™

BSplineMfd
0 ⎧ 1 u
N i ( u)
i
≤ u<
u
= ⎨
i + 1
⎩ 0 otherwise
p u–
u i p – 1 u
N i ( u)
----------------------- i + p + 1
– u p – 1
= N
– i
( u)
+ -------------------------------------- N
– i + 1
( u)
u i + p
u i
A general B-spline can be described by
u i + p + 1
u i + 1
S( uv , )
=
n
∑
m
∑
i = 0 j = 0
n m
∑
∑
i = 0 j = 0
p q
N i ( u)Nj
( v)wi
, j
b i,
j
------------------------------------------------------------------------- , a ≤u≤
b,
c≤
v≤
d
p q
N i ( u)Nj
( v)wi
, j
where
U = { a , … , au , , p + 1
… , u , m – p – 1
b , … , b}
and
V = { c , … , cv , , p + 1
… , v , m – p – 1
d , … , d}
are the two knot vectors stored in the entry, and b and w are the control points
coordinates and weights stored in P.
For periodic splines, the first and last parameter values in the knot vectors are not
duplicated.
438 | CHAPTER 3: THE COMSOL MULTIPHYSICS FILES